Let $mathbf{x}_{j}$ be the $j$ th row of $mathbf{X}$, and $mathbf{X}_{-j}$ be the $mathbf{X}$ matrix with the $j$ th row removed. Show that [operatorname{Var}left[hat{beta}_{j} ight]=sigma^{2}left[mathbf{x}_{j}^{prime}

Let $\mathbf{x}_{j}$ be the $j$ th row of $\mathbf{X}$, and $\mathbf{X}_{-j}$ be the $\mathbf{X}$ matrix with the $j$ th row removed. Show that

\[\operatorname{Var}\left[\hat{\beta}_{j}\right]=\sigma^{2}\left[\mathbf{x}_{j}^{\prime} \mathbf{x}_{j}-\mathbf{x}_{j}^{\prime} \mathbf{X}_{-j}\left(\mathbf{X}_{-j}^{\prime} \mathbf{X}_{-j}\right)^{-1} \mathbf{X}_{-j}^{\prime} \mathbf{x}_{j}\right]\]